WASHINGTON (Reuters) -Acting NASA chief Sean Duffy and billionaire Elon Musk on Tuesday traded barbs online over who should lead the space agency, a day after Duffy invited companies to compete with Musk's SpaceX for a moon landing mission.

The spat involving the U.S. space program and the SpaceX CEO, which have extensive financial ties, spilled into public view after Musk responded on social media platform X to reports Duffy wants to fold the National Aeronautics and Space Administration into the Transportation Department. Duffy is also the Transportation Secretary.

"The person responsible for America's space program can't have a 2 digit IQ," Musk wrote.

The day before, Duffy said on Fox News' "Fox & Friends" program that development of SpaceX's Starship is behind schedule in its mission to return humans to the moon under the agency's Artemis program, an effort rivaling China's moon program.

As a result, Duffy said, NASA will invite other companies to compete for the mission, which was awarded to SpaceX in 2021.

A NASA spokeswoman said Duffy is focused on beating China to the Moon.

"Sean said that NASA might benefit from being part of the Cabinet, maybe even within the Department of Transportation, but he's never said he wants to keep the job himself," she added, referring to a Wall Street Journal report stating that such a move could secure Duffy's role at NASA.

Musk and SpaceX did not respond to requests for comment.

"Love the passion. The race to the Moon is ON," Duffy said on X on Tuesday, replying to a Musk post that bet Starship will "end up doing the whole Moon mission."

"Great companies shouldn't be afraid of a challenge," Duffy added.